This paper presents a unified framework for dynamic graph compression with applications to the multicut problem in planar graphs where all terminals lie on the outer face. We develop an on-the-fly algorithm for computing and maintaining the maximum flow minimum cut gap in dynamically changing networks, achieving O(log n) update time for single-edge modifications. For planar graphs with k terminals on the outer boundary, we establish that the multicut problem admits an exact polynomial-time solution and provide an O(n log n) algorithm based on shortest-path separators. We extend these results to handle deep emulations of fault-free mesh architectures on meshes with random faults, proving an O(log n) slowdown bound for constant fault probability. Additionally, we develop compression techniques for adjacency list representations tailored to web graphs, mobile host networks, and social network structures, achieving 2.1 bits per edge for scale-free graphs. The algorithms are validated on real-world datasets including ClueWeb09, Twitter's follower graph, and mobile network traces from the CRAWDAD repository.